Steinhaus观点下的正规数
Normal numbers from Steinhaus viewpoint
摘要：
本文回顾了文献[0，1]中Borel σ-代数B的一个非标准构造，并利用完全非拓扑方法构造了B中的测度族（特别是勒贝格测度）.这种方法可以追溯到1923年的Steinhaus，现在用于引入正常数概念的自然推广，并探索其内在的概率性质。我们证明，在几乎所有的情况下，几乎所有的实数在[0，1]是正常的（关于这个广义的概念）。这个过程突出了[0，1]中Borel σ-代数和勒贝格测度的一些明显隐藏但有趣的特征。
英文摘要：
In this paper we recall a non-standard construction of the Borel sigma-algebra B in [0,1] and construct a family of measures (in particular, Lebesgue measure) in B by a completely non-topological method. This approach, that goes back to Steinhaus, in 1923, is now used to introduce natural generalizations of the concept of normal numbers and explore their intrinsic probabilistic properties. We show that, in virtually all the cases, almost all real number in [0,1] is normal (with respect to this generalized concept). This procedure highlights some apparently hidden but interesting features of the Borel sigma-algebra and Lebesgue measure in [0,1].
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